vault backup: 2026-04-08 10:52:50
This commit is contained in:
Jan Meyer
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---
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created: 2026-04-08 08:52
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course: "[[29593850 - Automationtheory]]"
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topic: strings
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related: "[[29593929 - Alphabets]]"
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type: lecture
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status: 🔴
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tags:
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- university
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---
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## 📌 Summary
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> [!abstract]
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> Overview of lecture 1 on `Wednesday, 2026/Apr/08`
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---
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## 📝 Content
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A word (or _string_) is a finite sequence $w = a_1 a_2 ... a_n$ if characters from $Sigma$.
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> [!CONVENTION]
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> We will use small letters to describe strings that are part of a language.
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> [!EXAMPLE]
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> $"aa", "ab", "bba"$ and $"baab"$ are strings over $Sigma = {a, b}.
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#### Length of a string
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The _length_ $abs(x)$ of a string $x = a_1 ... a_n$ is its number $abs(x) = n$ of characters.
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#### Empty String
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The empty string is denoted by $epsilon$, this is the neutral element.
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-> $abs(epsilon) = 0$
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## String Operations
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### Concatenation
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String can be concatenated, where one string is appended to another.
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For strings $x = a_1 ... a_n$ and $y = b_1 ... b_m$ over alphabets $Sigma_x$ and $Sigma_y$, their _concatenation_ over the alphabet $Sigma = Sigma_x union Sigma_y$ is the string
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$$x circle.small y = x y = a_1 a_2 ... a_n b_1 b_2 ... b_m$$
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> This string is of the length $abs(x y) = n + m$
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> [!EXAMPLE]
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> $x = "apple"$
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> $y = "pie"$
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> $x circle.small y = "applepie"$
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Order of operations / Brackets do _not matter_. (Concatenation is associative but **not** commutative $x y eq.not y x$)
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> $(x circle.small y) circle.small z = x circle.small (y circle.small z)$
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Any string concatenated with the empty string $epsilon$ will result in itself.
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> $x circle.small epsilon = x = epsilon circle.small x$
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### Exponentiation
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The $n^"th"$ power $x^n$ of a string $x$ is the $(n-1)$-fold concatenation of $x$ with itself.
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> $x^0 := epsilon$
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> $x^n := x^(n-1) circle.small x$ for $n in NN$
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> [!Example]
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> $x^4 = x x x x$
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> $(a b)^3 = a b a b a b$
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### Reversing / Mirroring
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For a string $x = a_1 a_2 ... a_(n-1) a_n$ of length $n$, it's _mirrored string_ is given by
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$$ x^("Rev") = a_n a_(n-1)...a_2 a_1$$
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## Substrings
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A string $x$ is a _substring_ of a string $y$ if $y = u x v$, where $u$ and $v$ can be arbitrary strings.
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- If $u = epsilon$ then $x$ is a _prefix_ of $y$.
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- If $v = epsilon$ then $x$ is a suffix of $y$.
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For strings $x$ and $y$ the quantity $abs(y)_x$ is the number of times that $x$ is a substring of $y$.
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@@ -1,26 +0,0 @@
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---
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created: 2026-04-08 10:09
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course: "[[29593850 - Automationtheory]]"
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topic: alphabets, characters
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related:
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type: lecture
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status: 🟢
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tags:
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- university
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---
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## 📌 Summary
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> [!abstract]
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> Definition and examples of alphabets.
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---
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## 📝 Content
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Alphabets are formal, non-empty, finite, sets of characters (or _letters_ or _symbols_). They are denoted by $Sigma$.
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$Sigma = {a, b}$
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> Alphabet $Sigma$ contains the characters $a$ and $b$.
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$Sigma = {a, ..., z, A, ..., Z, 0, ..., 9}$
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> usual alphabet for writing text
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@@ -1,46 +0,0 @@
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---
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created: 2026-04-08 10:15
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course: "[[29593850 - Automationtheory]]"
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topic: kleen
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related: "[[29593929 - Alphabets]]"
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type: lecture
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status: 🟢
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tags:
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- university
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---
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## 📌 Summary
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> [!abstract]
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>
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---
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## 📝 Content
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### Kleene Star
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Denoted by $Sigma^*$. The Kleene Star (or _Kleene operator_ or _Kleene Closure_) gives an infinite amount of strings made up of the characters of the alphabet $Sigma ^ *$.
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$Sigma^*$ is the set of all string that can be generated by arbitrary concatenation of its characters.
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> $Sigma^* := union.big_(n>=0) A_n$
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> where $A_n$ is the set of all string combinations of length $n$
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#### Remarks
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- The same character can be used multiple times.
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- The empty string $epsilon$ is also part f $Sigma^*$.
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> [!Example]
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> $Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}$
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> [!FACT]
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> - The set $Sigma^*$ is infinite, since we defined $Sigma$ to be non-empty.
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> - It is _countable_ and has the same cardinality as the set $NN$ of natural numbers
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### Kleene Plus
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The _Kleene Plus_ of an alphabet $Sigma$ is given by $Sigma^+ = Sigma^* backslash {epsilon}$
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### Lemma group structure
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The structure _Lemma_ is induced by the Kleene star - it is a monoid, that is a semigroup with a neutral element.
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> [!PROOF]
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> - Associativity has been shown
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> - Existence of a neutral element has been shown.
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> - Closure under $circle.small$: Let $x in Sigma^*$ and $y in Sigma^*$ be two string over the alphabet $Sigma$. Then $x circle.small y = x y in Sigma^*$
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@@ -1,58 +0,0 @@
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---
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created: 2026-04-08 10:20
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course: "[[29593850 - Automationtheory]]"
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topic: languages
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related:
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type: lecture
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status: 🔴
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tags:
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- university
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---
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## 📌 Summary
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> [!abstract]
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> Definition and example for formal languages
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---
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## 📝 Content
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A formal _language_ of the alphabet $Sigma$ is a subset $L$ of $Sigma^*$.
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> [!EXAMPLE]
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> For alphabet $Sigma = {a, b}$, let $L_1$ be the set of all string starting with $b$, followed by an arbitrary number of $a$'s, and ending with $b$:
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> $L_1 = {b a^n b | n in NN_0}$
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> Then $b b in L_1, b a b in L_1$, etc.
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>
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>More in [[29593895 - atfl-st2026-l01-formal-languages-full.pdf#page=54]]
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## Language Operations
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### Concatenation of languages
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For languages $X subset Sigma^*_X$ over alphabet $Sigma_X$ and $Y subset Sigma^*_Y$ over alphabet $Sigma_Y$, their _concatenation_ is
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> $X circle.small Y = X Y = {x y bar x in X and y in Y}$
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The concatenation of $X$ and $Y$ thus contains all string combinations where the prefix is a string from $X$ and the suffix is a string from $Y$.
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> [!CONVENTION]
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> Concatenation has a higher precedence than set operations ($union, inter$).
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### Exponentiation of languages
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The $n^"th"$ power of the language $L subset.eq Sigma^*$ over alphabet $Sigma$ is
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- $L^0 := { epsilon }$
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- $L^n := L^(n-1) L$ if $n > 0$
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## Finite representation of languages
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**Goal:** Represent a language using _finite_ information
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### Using set notation
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$S = {a^n b^m bar n, m >= 0} = {epsilon, a, b, "aa", "ab", ...}$
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> This is very inefficient.
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### Using regular expressions
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A _regular expression_ $r$ over an alphabet $Sigma$ is defined recursively:
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- $emptyset, epsilon$ and each $a in Sigma$ are regular expression, which represent the Languages $L(emptyset) = emptyset, L(epsilon) = {epsilon}$ and $L(a) = {a}$
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- If $r$ and $s$ are regular expressions then
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- $(r+s)$
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@@ -1,20 +0,0 @@
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---
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created: 2026-04-08 10:32
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course: "[[29593850 - Automationtheory]]"
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topic: languages
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related: "[[29593940 - Formal Languages]]"
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type: lecture
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status: 🔴
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tags:
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- university
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---
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## 📌 Summary
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> [!abstract]
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>
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---
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## 📝 Content
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A language $L$ that can be described by a regular expression $r$ (i. e. $L(r) = L$) is called _regular_.
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@@ -1,25 +0,0 @@
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---
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created: 2026-04-08 10:38
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course: "[[29593850 - Automationtheory]]"
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topic: kleene
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related: "[[29593940 - Formal Languages]]"
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type: lecture
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status: 🔴
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tags:
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- university
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---
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## 📌 Summary
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> [!abstract]
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>
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---
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## 📝 Content
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The _Kleene star_ $L^*$ of a language $L subset Sigma^*$ is the set of all strings (including the empty string $epsilon$) that can be generated by arbitrary concatenation of strings in the language, that is
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> $L^* := union.big_(n >= 0) L^n$
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> [!EXAMPLE]
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> For $L = {01}$ one has $L^* = {epsilon, 01, 0101, 010101, ...}$
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> $= {(01)^n bar n >= 0}$
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